Pegging resources to shipments in a pre-existing production plan

ABSTRACT

In software systems for resource-constrained production planning (RCPP), it is often beneficial to determine a detailed association, or “pegging”, between the resources and the shipments that are enabled by those resources. Presented here is a software-implemented method for constructing a pegging of a production plan for an RCPP problem. The method is called “Pegging By Explosion” (PBE). This method, takes a feasible production plan and reconstructs it, using its own specialized incremental allocation technique. A Pegging By Monitoring technique is applied to this reconstruction in order to produce the pegging. The reconstruction is accomplished by a specialized “explosion” technique related to the explosion technique of Material Requirements Planning (MRP).

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to manufacturing resourceplaning, and more particularly to software systems forresource-constrained production planning.

2. Background Description

Software systems for resource-constrained production planning (RCPP)take as input data a description of a production planning problem. Thisincludes a listing of the basic data objects of the problem; i.e.,capacity resources, raw material resources, subassemblies, finalproducts, and demands for the products, as well as a multi-levelBill-Of-Resources network that defines how the products can be producedfrom subassemblies, raw materials and capacities. It also includesvarious attributes on these objects such as supply volumes, demandvolumes, production usage rates, and economic data or priorities. Inaddition, the problem data is defined with respect to a set of discretetime periods.

Given these inputs, an RCPP system computes a production plan thatconsists of two portions: (1) a production portion of plan thatspecifies of how much of each product is to be produced in each timeperiod, and (2) a shipment portion of the plan that specifies how muchof each demand is to be fulfilled in each time period. (The fulfillmentof a demand in a period may be called a “shipment” to that demand.) Theproduction plan computed by the RCPP system is normally required to befeasible, meaning it satisfies the resource constraints that arespecified in the input data.

Two methods for solving the RCPP problem are described in U.S. Pat. No.5,548,518 for “Allocation Method for Generating a Production Schedule”and U.S. Pat. No. 5,630,070 for “Optimization of Manufacturing ResourcePlanning”, both to Dietrich et al. More specifically, U.S. Pat. No.5,548,518 discloses an allocation method which, in response to aspecified requirement, determines what quantity of a product can beproduced with a specified quantity of supply components, allocates arequired quantity of supply components for filling a defined partialorder and fills a remainder of the requirement at some later time. U.S.Pat. No. 5,630,070 discloses a method for constrained materialrequirements planning, optimal resource allocation and productionplanning which optimizes a manufacturing process by designating theamounts of various manufactured products to be produced, which productsinclude both end products as well as subassemblies.

Users of RCPP systems frequently desire more information than just theproduction plan itself. One question that is often asked is, “For anygiven shipment in the production plan, which resources were utilizedspecifically to enable that shipment?” Generally, this information isnot evident in the production plan itself. The production plan specifieswhich resources are used and which demands are met, but it does notprovide an association between the two. This may be called “the peggingproblem” for RCPP; that is, given an RCPP problem and a correspondingfeasible production plan, define an association between the individualresources that are utilized in the plan and the individual shipments inthe plan. Such an association is called a “pegging” of the productionplan. A resource that is associated with a particular shipment is saidto be “pegged” to it.

Users of RCPP systems have many uses for a pegging of a production plan,including the following:

-   -   Verifying the validity of the production plan.    -   Verifying the validity of the input data.    -   Identifying which demands are responsible for depleting the        supplies of critical (i.e., bottleneck) resources.    -   Identifying opportunities to improve the production plan by        increasing the supplies of the resources that were used by the        most important demands.    -   Generally deepening the user's understanding of the production        plan.

One possible method for computing a pegging might be called “Pegging ByMonitoring”. This technique would apply when the RCPP system solves theRCPP problem by an “incremental allocation” technique, as disclosed forexample in U.S. Pat. No. 5,548,518. An incremental allocation techniquefor RCPP has the following key properties:

-   -   It starts with an empty production plan.    -   It proceeds by iterations.    -   Each iteration creates only one new shipment.    -   As the technique progresses, the production volume for any        product in any period can be increased, but it can never be        decreased.        The “Pegging By Monitoring” technique would proceed by        monitoring the incremental allocation technique; that is,        whenever a production volume of a product is increased by the        incremental technique, that increase is pegged to the shipment        for the current iteration.

In some respects, “Pegging By Monitoring” is a very appealing form ofpegging. It s a direct reflection of what how the production plan wasbuilt up by the incremental allocation technique. But it has thefollowing drawbacks:

-   -   It can only be applied, if the production plan is being computed        by an incremental allocation technique.    -   It must be integrated with the RCPP solver itself; it cannot be        applied to a production plan that has already been computed.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide a methodfor solving the pegging problem in a resource-constrained productionplanning system which avoids the problems of the “Pegging By Monitoring”approach.

According to the invention, there is provided a method for constructinga pegging of a production plan for the RCPP problem which is calledherein “Pegging By Explosion” (PBE). The Pegging By Explosion techniquetakes a feasible production plan and reconstructs it, using its ownspecialized incremental allocation technique. The Pegging By Monitoringtechnique (described above) is applied to this reconstruction in orderto produce the pegging. The reconstruction is accomplished by aspecialized “explosion” technique related to the well-known explosiontechnique of Material Requirements Planning (MRP). (For details aboveMRP, see Manufacturing Planning and Control Systems, by T. E. Vollmann,W. L. Berry, and D. C. Whybark, McGraw-Hill, 4th ed., Mar. 1, 1997). Aconventional MRP explosion technique would usually not be able toreconstruct an arbitrary production plan to an RCPP problem. Incontrast, the explosion technique used by PBE does so, by making use ofdetailed knowledge of the production plan itself.

The advantages of using the Pegging By Explosion technique instead ofthe Pegging By Monitoring technique are as follows:

-   -   It can be used regardless of how the production plan was        generated.    -   It does not need to be integrated with the RCPP solver.        Note that a pegging of a production plan is not uniquely        determined by the production plan itself. For any given feasible        production plan, there will normally be many different possible        peggings. In light of this, it is important to construct a        pegging that a user is likely to find “plausible”. Specifically,        it should be plausible that the resources that are pegged to a        particular shipment were precisely those resources that made        that shipment possible. The PBE technique of the present        invention constructs a pegging that is guaranteed to satisfy        this “plausibility” condition.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects and advantages will be betterunderstood from the following detailed description of a preferredembodiment of the invention with reference to the drawings, in which:

FIG. 1 is a data flow diagram illustrating the PBE technique in thecontext of an RCPP system;

FIG. 2 is a data flow diagram similar to FIG. 1 with the addition of ashipment sequence;

FIG. 3 is a flow diagram illustrating the logic of the PBE algorithmaccording to the invention;

FIG. 4 is a flow diagram of the procedure “Explode-Triple” used in thealgorithm of FIG. 3; and

FIG. 5 is a flow diagram of the procedure “Main-Iteration” used in theprocedure of FIG. 4.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

The invention disclosed here is a novel method for constructing apegging of a production plan for an RCPP problem. The technique iscalled “Pegging By Explosion” (PBE).

The Pegging By Explosion technique takes a feasible production plan andreconstructs it, using its own specialized incremental allocationtechnique. The Pegging By Monitoring technique (described above) isapplied to this reconstruction in order to produce the pegging. Thereconstruction is accomplished by a specialized “explosion” techniquerelated to the well-known explosion technique of Material RequirementsPlanning (MRP). (For details above MRP, see Manufacturing Planning andControl Systems, by T. E. Vollman, W. L. Berry, and D.C. Whybark.) Aconventional MRP explosion technique would usually not be able toreconstruct an arbitrary production plan to an RCPP problem. Incontrast, the explosion technique used by PBE does so by making use ofdetailed knowledge of the production plan itself.

The data flow for the PBE technique in the context of an RCPP system isillustrated in FIG. 1. As shown in FIG. 1, the RCPP problem data 1 isinput to the RCPP solver 2, which produces a production plan 3. ThePegging by Explosion (PBE) technique 4 according to the presentinvention is applied to the thus produced production plan 3, yieldingthe pegging 5 of the production plan.

The advantages of using the Pegging By Explosion technique instead ofthe Pegging By Monitoring technique are as follows:

-   -   It can be used regardless of how the production plan was        generated.    -   It does not need to be integrated with the RCPP solver.        Note that a pegging of a production plan is not uniquely        determined by the production plan itself: For any given feasible        production plan, there will normally be many different possible        peggings. In light of this, it is important to construct a        pegging that a user is likely to find “plausible”. Specifically,        it should be plausible that the resources that are pegged to a        particular shipment were precisely those resources that made        that shipment possible. Hereinafter, a precise mathematical        condition will be defined that formalizes of the idea of a        plausible pegging. The PBE technique constructs a pegging that        is guaranteed to satisfy this “plausibility” condition.

This is an algorithmic invention. It would be implemented in computersoftware, normally as an additional capability of an RCPP system,although in principle, it could be implemented as a stand-alone softwaretool.

Before describing the PBE technique, the underlying RCPP problem must bedefined.

The RCPP Problem

An RCPP problem is built up out of several kinds of data objects:

-   Periods: A period represents a distinct interval of time, such as a    day, a week, two months, a year, etc. The periods in an RCPP problem    need not be of uniform duration. The periods are represented by the    integers from 1 to Nperiods, the number of periods.-   Resources: These represent capacity resources, raw material    resources, subassembly resources, and even the final products    themselves.-   Materials: These are resources that remain available in the next    period, if they are not used in a given period.-   Capacities: These are resources that do not remain available in the    next period, if they are not used in a given period. Each resource    is either a material or a capacity, but not both.-   Products: These are the resources that can be produced. They may be    either materials or capacities.-   Operations: An operation represents a means by which a particular    product can be produced. Each operation produces exactly one    product; each product is produced by one or more operations. If a    product is produced by more than one operation, this simply    indicates that there is more than one way to produce that product.-   Demands: Each demand is associated with a resource and represents    demand for that resource. Each resource is associated with zero or    more demands.

Let

-   -   AllPer=The set of all periods={1, 2, . . . , Nperiods}    -   AllRes=The set of all resources    -   AllMat=The set of all materials (Thus, AllMat        AllRes.)    -   AllCap=The set of all capacities (Thus, AllCap        AllRes.)    -   AllOpn=The set of all operations    -   AllDem=The set of all demands        ∀dεAllDem:    -   Let DemRes (d)=The resource demanded by demand d.        ∀rεAllRes:    -   Let ResDems (r)={dεAllDem|r=DemRes (d)}        ∀jεAllOpn:    -   Let ProdRes (j)=The resource produced by operation j.        ∀rε AllRes:    -   Let ProdOpns (r)={jεAllOpn|r=ProdRes (j)}        -   [These are the “producing operations” for resource r.]        -   [Resource r is a product, if and only if ProdOpns (r)≠Ø.]            ∀jε AllOpn:    -   Let BOR (j)=The set of all resources that must be consumed by        operation j in order to produce ProdRes (j). This is the        “Bill-Of-Resources” of operation j.        ∀r□ AllRes:    -   Let ConsOpns (r)={jεAllOpn|rεBOR(j)}        -   [These are the “consuming operations” of resource r.]

Given any resource r and any period tr, there is an associatednonnegative quantity, SupplyVol (r, tr) (“supply volume”), thatspecifies the amount of resource r that is available as supply in periodtr. Given any demand d and any period td, there is an associatednonnegative quantity, DemandVol (d, td) (“demand volume”), thatspecifies the amount of resource DemRes (d) that is demanded by demand din period td.

Given any operation j, any period tj, any resource rεBOR (i), and anyperiod tr, there is an associated quantity, UsageRate (i, tj, r, tr)(“usage rate”), that specifies the amount of resource r that must beconsumed in period tr for each unit of ProdRes (i) that is produced byoperation j in period tj. Note that UsageRate (i, tj, r, tr) may benegative. This indicates that, when operation j produces ProdRes (j) inperiod tj, it consumes a negative amount of resource r in period tr.This can be used to model operations that have two or more outputs:ProdRes (j) is the main product, while resources rεBOR (j) withUsageRate (j, tj, r, tr)<0 are by-products. Many MRP databases containnegative usage rates in their BOR data.

With each resource r, there is an associated integer, ResIdx (r)(“resource index”), distinct for each resource. This index defines anordering for the resources and has the following property:

∀jεAllOpn, tjεAllPer, ∀rεBOR (j), trεAllPer:

If UsageRate (j, tj, r, tr)>0, then:

-   -   ResIdx (ProdRes (j))>ResIdx (r).        Finally, there would also be economic or priority data        associated with the resources, operations, and demands, but this        data is not necessary for defining a pegging technique.

A solution to an RCPP problem is a production plan. Given any operation,j, and any period, tj, the production plan specifies an associatedquantity, ProdVol (j, tj) (“production volume”) that represents theamount of resource ProdRes (j) that is to be produced by operation j inperiod tj. Given any resource, r, and any period, tr, the productionplan specifies an associated quantity, StockVol (r, tr) (“stock volume”)that represents the amount of resource r that is to be carried over fromperiod tr to period tr+1. If r is a capacity, then StockVol (r, tr)=0.For notational convenience, let StockVol (r, 0)=0. Given any resource,r, and any period, tr, the production plan specifies an associatedquantity, ConsSupVol (r, tr) (“consumed supply volume”) that representsthe amount of the supply of resource r from period tr that is consumedin any period. Given any demand, d and any period td, the productionplan specifies an associated quantity, ShipVol (d, td) (“shipmentvolume”) that represents the quantity of the demand that is to befulfilled (i.e., “shipped”) in period td. The collection of productionvolumes of all operations, stock volumes and consumed supply volumes ofall resources, and shipment volumes of all demands constitutes theproduction plan.

Given a production plan, the following further quantities may bedefined:

∀r ∈ AllRes, ∀tr ∈ AllPer:$\mspace{79mu} {{{Let}\mspace{14mu} {{TotShipVol}\left( {r,{tr}} \right)}} = {\sum\limits_{{{{d \in \; {ResDems}})}r})}\; {{{ShipVol}\left( {d,{tr}} \right)}{\quad \mspace{79mu} {{{Let}\mspace{14mu} {{TotProdVol}\left( {r,{tr}} \right)}} = {\sum\limits_{j \in \; {{ProdOpns}{(r)}}}\; {{{ProdVol}\left( {j,{tr}} \right)}{\quad \mspace{79mu} {{{Let}\mspace{14mu} {ConsVol}\left( {r,{tr}} \right)} = {\sum\limits_{\substack{j \in \; {{ConsOpns}{(r)}} \\ y \in \; {AllPers}}}{{{UsageRate}\left( {j,{tj},r,{tr}} \right)} \cdot {{ProdVol}\left( {j,{tj}} \right)}}}}}}}}}}}}$

TotShipVol (r, tr) is the total amount of resource r that is shipped inperiod tr (“total shipment volume”). TotProdVol (r, tr) is the totalamount of resource r that is produced in period tr (“total productionvolume”). ConsVol (r, tr) is the amount of resource r that is consumedin period tr (“consumption volume”). Note that, if r is not a product,then TotProdVol (r, tr)=0. A production plan is required to satisfy thefollowing constraints:

  ∀j ∈ AllOpn, ∀tj ∈ AllPer:  ProdVol  (j, tj) ≥ 0[non-negativity  constraints]  ∀d ∈ AllDem, ∀td ∈ AllPer:  ShipVol  (d, td) ≥ 0[non-negativity  constraints]  ∀r ∈ AllMat, ∀tr ∈ AllPer:  StockVol  (r, tr) ≥ 0[non-negativity  constraints]  ∀r ∈ AllCap, ∀tr ∈ AllPer:  StockVol  (r, tr) = 0 [non-stocking  constraints]  ∀r ∈ AllRes, ∀tr ∈ AllPer:  ConsSupVol  (r, tr) ≥ 0 [non-negativity  constraints]  ∀r ∈ AllRes, ∀tr ∈ AllPer:$\mspace{20mu} {{ConsSupVol}\mspace{11mu} {\left( {r,{tr}} \right)\;\left\lbrack {{{SupplyVol}\mspace{11mu} {\left( {r,{tr}} \right)\left\lbrack {{supply}\mspace{14mu} {constraints}} \right\rbrack}\mspace{20mu} {\forall{d \in {AllDem}}}},{\forall{{td} \in {{{AllPer}\text{:}{\sum\limits_{{td}^{\prime} = 1}^{td}{{ShipVol}\mspace{11mu} \left( {d,{td}^{\prime}} \right)}}} \leq {\sum\limits_{{td}^{\prime} = 1}^{td}{{DemandVol}\mspace{11mu} {\left( {d,{td}^{\prime}} \right)\left\lbrack {{demand}\mspace{14mu} {contraints}} \right\rbrack}\mspace{20mu} {\forall{r \in {AllRes}}}}}}}},{\forall{{tr} \in {{{{AllPer}\text{:}{TotShipVol}\mspace{11mu} \left( {r,{tr}} \right)} + {{ConsVol}\mspace{11mu} \left( {r,{tr}} \right)} + {{StockVol}\mspace{11mu} \left( {r,{tr}} \right)}} \leq {{{ConsSupVol}\mspace{11mu} \left( {r,{tr}} \right)} + {{TotProdVol}\mspace{11mu} \left( {r,{tr}} \right)} + {{StockVol}\mspace{11mu} {\left( {r,{{tr} - 1}} \right)\mspace{11mu}\left\lbrack {{resource}\mspace{14mu} {contraints}} \right\rbrack}}}}}}} \right.}}$

A production plan that satisfies all of the above constraints is said tobe feasible.

An important special case of the RCPP problem is the PositiveConsumption RCPP problem. An RCPP problem is defined to be a PositiveConsumption RCPP (PC RCPP) problem, if it has the following property:

∀jεAllOpn, tjεAllPer, ∀rεBOR (j), tr e AllPer:

UsageRate (i, tj, r, tr)≧0

Note that this implies the following property:∀jεAllOpn, ∀rεBOR (j):

ResIdx (ProdRes (j))>ResIdx (r).

Precise Definition of Pegging

Given an RCPP problem and feasible production plan for it, a pegging ofthe production plan has three portions: (1) a production pegging, (2) astock pegging, and a (3) supply pegging. Given any operation j, anyperiod tj, any demand d, and any period td, a production peggingspecifies an associated quantity, PgdProdVol (j, tj, d, td) (“peggedproduction volume”). This is the portion of ProdVol (j, tj) that ispegged to the shipment of demand d in period td.

A production pegging must satisfy the following two constraints:

∀j ∈ AllOpn, ∀tj ∈ AllPer, ∀d ∈ AllDem, ∀td ∈ AllPer:       PgdProdVol(j, tj, d, td) ≥ 0[non-negativity  constraints]${\forall{j \in {AllOpn}}}, {\forall{{tj} \in {{AllPer}\text{:}{\quad\mspace{14mu} \mspace{110mu} {\quad{{\sum\limits_{\underset{{td} \in \; {AllPers}}{d \in \; {AllDems}}}\; {{PgdProdVol}\left( {j, {tj}, d, {td}} \right)}} \leq {{{ProdVol}\left( {j, {tj}} \right)}{\quad\left\lbrack {{total}\mspace{14mu} {pegging}\mspace{14mu} {constraints}} \right\rbrack}}}}}}}}$

Similarly, given any resource r, any period tr, any demand d, and anyperiod td, a stock pegging specifies an associated quantity, PgdStockVol(r, tr, d, td) (“pegged stock volume”). This is the portion of StockVol(r, tr) that is pegged to the shipment of demand d in period td.

A stock pegging must satisfy the following two constraints:

∀r ∈ AllRes, ∀tr ∈ AllPer, ∀d ∈ AllDem, ∀td ∈ AllPer: $\mspace{79mu} {{{{PgdStockVol}\left( {r,{tr},d,{td}} \right)} \geq {{0\left\lbrack {{non}\text{-}{negativity}\mspace{14mu} {constraints}} \right\rbrack}{\forall{r \in {AllRes}}}}},{\forall{{tr} \in {{AllPer}\text{:}{\quad \mspace{124mu} {\quad{{\sum\limits_{\underset{{td} \in \; {AllPers}}{d \in \; {SllDems}}}\; {{PgdStockVol}\left( {r, {tr}, d, {td}} \right)}} \leq {{{StockVol}\left( {r,{tr}} \right)}{\quad\left\lbrack {{total}\mspace{14mu} {pegging}\mspace{14mu} {constraints}} \right\rbrack}}}}}}}}}$

Note that this implies that PgdStockVol (r, tr, d, td)=0, if r is acapacity.

Finally, given any resource r, any period tr, any demand d, and anyperiod td, a supply pegging specifies an associated quantity,PgdConsSupVol (r, tr, d, td) (“pegged consumed supply volume”). This isthe portion of ConsSupVol (r, tr) that is pegged to the shipment ofdemand d in period td.

A supply pegging must satisfy the following two constraints:

∀r ∈ AllRes, ∀tr ∈ AllPer, ∀d ∈ AllDem, ∀td ∈ AllPer:       PgdConsSupVol(r, tr, d, td) ≥ 0[non-negativity  constraints]${\forall{r \in {AllRes}}},{\forall{{tr} \in {{AllPer}\text{:}{\quad \mspace{124mu} {\quad{{\sum\limits_{\underset{{td} \in \; {AllPers}}{d \in \; {AllDems}}}\; {{PgdConsSupVol}\left( {r,{tr},d, {td}} \right)}} \leq {{{ConsSupVol}\left( {r,{tr}} \right)}{\quad\left\lbrack {{total}\mspace{14mu} {pegging}\mspace{14mu} {constraints}} \right\rbrack}}}}}}}}$

A pegging of a production plan for an RCPP problem is simply aproduction pegging, stock pegging, and supply pegging that satisfy theabove six constraints. The total pegging constraints enforce arequirement that any quantity of production, stock, or supply that ispegged to one shipment cannot be pegged to any other. Note that thetotal pegging constraints are inequality constraints. This reflects thefact that it is acceptable for some portion of the production, stock,and supply not to be pegged to any shipment.

Plausible Peggings The “Feasible Partitioning” Property

Given an RCPP problem, a feasible production plan for it, and a peggingof the production plan, and given any demand, d* and any period, td*,the pegged production plan for (d*, td*) is defined as follows:

-   -   ShipVol^(P) (d*, td*)=ShipVol (d*, td*)    -   ∀dεAllDem, ∀tdεAllPer:        -   If d≠d* or td≠td*, then ShipVol^(P) (d, td)=0    -   ∀j▴AllOpn, ∀tjεAllPer:        -   ProdVol^(P) (i, tj)=PgdProdVol (i, tj, d*, td*)    -   ∀rεAllRes, ∀trεAllPer:    -   StockVol^(P) (r, tr)=PgdStockVol (r, tr, d*, td*)    -   ConsSupVol^(P) (r, tr)=PgdConsSupVol (r, tr, d*, td*)        The pegging is defined to be a feasible partitioning of the        production plan if, for each demand d* and each period td*, the        pegged production plan for (d*, td*) is feasible. Informally,        this means that the pegging has partitioned the production plan        into a set of disjoint portions associated with each shipment,        where each portion defines a feasible production plan for        achieving its corresponding shipment. This is a precise way of        saying that the pegging is plausible.

In the context of a Positive Consumption RCPP problem, the peggingconstructed by the PBE technique is guaranteed to be a feasiblepartitioning.

The Shipment Sequence

In addition to an RCPP problem and a production plan, the PBE techniqueoptionally accepts one extra item of input data: a shipment sequence,ShipSeq. This is an ordered list of triples (d, td, IncShipVol), where dC AllDem, tdεAllPer, and IncShipVol≧0 (“incremental shipment volume”).The shipment sequence is required to satisfy the following constraints:

${\forall{d \in {AllDem}}},{{\forall{{td} \in {{AllPer}\text{:}\mspace{11mu} {\sum\limits_{{({d,\; {td},\; {IncShipVol}})} \in \; {ShipSeq}}\; {IncShipVol}}}}} = {{ShipVol}\left( {d,{td}} \right)}}$

In essence, a shipment sequence is simply the set of all shipments inthe production plan arranged into some specific order, where theshipments may be broken into partial shipments that can be interlacedwithin the sequence.

The shipment sequence is optional input, because the PBE technique caneasily generate one by default. For a default shipment sequence. it cansimply go through the periods in order, and for each period, it can gothrough the demands in any order:

{(demand #1, period #1, shipVol (demand #1, period #1)),(demand #2, period #1, shipVol (demand #2, period #1)),. . . ,(demand #1, period #2, shipVol (demand #1, period #2)),. . . }

By specifying a non-default shipment sequence, the user can exercisesome control over the pegging that will result: The resources that thePBE technique pegs first will be pegged to the shipments that appearearliest in the shipment sequence.

The data flow for the PBE technique with an optional shipment sequenceis illustrated in FIG. 2. Here, blocks 6 to 10 correspond to blocks 1 to5 of FIG. 1, while block 11 shows how the optional shipment sequencerelates to the overall data flow.

The Pegging by Explosion Technique for the Positive Consumption RCPPProblem

The Pegging By Explosion technique will initially be described in thecontext of the PC RCPP problem. Following this initial description, thetechnique will be extended to apply to the fully general RCPP problem.

The PBE technique functions by reconstructing the production plan in asomewhat modified form. Let ProdVol^(R)( ), StockVol^(R)( ),ConsSupVol^(R)( ), and ShipVol^(R)( ) denote the reconstructedproduction plan. Upon conclusion of the reconstruction, the followingproperties hold:

-   -   ∀dεAllDem, ∀tdεAllPer:

ShipVol^(R)(d, td)=ShipVol(d, td)  (1)

-   -   ∀jεAllOpn, ∀tjεAllPer:

ProdVol^(R)(j, tj)≦ProdVol (j, tj)  (2)

-   -   ∀rεAllRes, ∀trεAllPer:

StockVol^(R)(r, tr)≦StockVol(r, tr)  (3)

ConsSupVol^(R)(r, tr)≦ConsSupVol(r, tr)  (4)

In addition, the reconstructed production plan is feasible. In otherwords, the reconstructed plan achieves the same set of shipments as theoriginal plan, by only using production, stock, and supply that was usedin the original.

The description of the PBE algorithm will be broken down into smaller“procedures”. The most important procedure in the PBE algorithm iscalled “Explode Triple”. This procedure takes as its argument a triple(d*, td*, IncShipVol) from the shipment sequence and computes anincremental production plan for the triple, denoted by ProdVol^(I)( ),StockVol^(I)( ), ConsSupVol^(I)( ), and ShipVol^(I)( ).

When the Explode Triple procedure returns, the following propertieshold:

-   -   ∀dεAllDem, ∀tdεAllPer:

If d=d* and td=td*, then ShipVol^(I)(d, td)=IncShipVol  (5)

Otherwise, ShipVol^(I)(d, td)=0  (5)

∀jεAllOpn, ∀tjεAllPer:

ProdVol^(R)(j, tj)+ProdVol^(I)(j, tj)≦ProdVol (j, tj)  (6)

-   -   ∀rεAllRes, ∀trεAllPer:

StockVol^(R)(r, tr)+StockVol^(I)(r, tr)≦StockVol (r, tr)  (7)

ConsSupVol^(R)(r, tr)+ConsSupVol^(I)(r, tr)≦ConsSupVol (r, tr)  (8)

The incremental production plan is feasible.  (9)

Note: In the following description, the symbol “←” representsassignment. Thus, “x←v” means “x is set to the value of y”. Also, thesymbol “←+” represents adding to, so that “x←+y” means “x←x+y”.

The top-level logic of the PBE technique can be described as follows:

Algorithm Pegging-By-Explosion:

[Initialize the reconstructed data and pegging data to zero]∀dεAllDem, ∀tdεAllPer, ∀jεAllOpn, ∀tjεAllPer, ∀rεAllRes, ∀trεAllPer:

ShipVol^(R)(d, td)←0

ProdVol^(R)(j,tj)←0

StockVol^(R)(r, tr)←0

ConsSupVol^(R)(r, tr)←0

PgdProdVol (j, tj, d, td)←0

PgdStockVol (r, tr, d, td)←0

PgdConsSupVol (r, tr, d, td)←0

For each triple (d*, td*, IncShipVol)εShipSeq, taken in sequence order,do the following:

Call Explode Triple (d*, td*, IncShipVol).

-   -   [Add the incremental production plan to the reconstructed        production plan.]

∀dεAllDem, ∀tdεAllPer:

-   -   ShipVol^(R)(d, td)←+ShipVol^(I)(d, td)

∀jεAllOpn, ∀tjεAllPer:

-   -   ProdVol^(R)(j, tj)←+ProdVol^(I)(j, tj)

∀rεAllRes, ∀trεAllPer:

-   -   StockVol^(R)(r, tr)←+StockVol^(I)(r, tr)    -   ConsSuPVol^(R)(r, tr)←+ConsSupVol^(I)(r, tr)    -   [Peg the incremental production, stock, and consumed supply        volumes to the current demand.]

∀jεAllOpn, ∀tjεAllPer:

-   -   PgdProdVol (j, tj, d*, td*)←+ProdVol^(I)(i, tj)

∀rεAllRes, ∀trεAllPer:

-   -   PgdStockVol (r, tr, d*, td*)←+StockVol^(I)(r, tr)    -   PgdConsSupVol (r, tr, d*, td*)←+ConsSupVol^(I)(r, tr)        Report the pegging: PgdProdVol ( ), PgdStockVol ( ),        PgdConsSupVol ( ). Stop.        Assuming that the Explode Triple procedure achieves conditions        (5), (6), (7), (8), and (9), it is easy to see that, upon        completion of the PBE technique, the reconstructed production        plan satisfies conditions (1), (2), (3), and (4) above and is        feasible. It is also easy to see that PgdProdVol( ),        PgdStockVol( ), and PgdConsSupVol( ) form a valid pegging of the        original production plan and that this pegging is a feasible        partitioning of the production plan. It remains to describe the        Explode Triple procedure.

FIG. 3 illustrates the top level logic of the PBE algorithm implementedin computer software. Upon starting the procedure at start block 12, aPC RCPP problem, a feasible production plan, and a shipment sequence,ShipSeq, are read in at function block 13. In initialization block 14,the reconstructed production is set to zero and pegging is set to zero.Then, processing block 15 for ∀(d*, td*, IncShipVol)εShipSeq in order isentered. This processing block iterates through the shipment sequence.The Explode-Triple (d*, td*, IncShipVol) process is called in functionblock 16. In function block 17, the incremental production plan is addedto the reconstructed production plan. Finally, the incrementalproduction plan is pegged to the shipment (d*, td*) in function block 18to complete the processing in processing block 15. Once this processingis complete, the pegging is reported in function block 19, and theprocess stops in stop block 20.

FIG. 4 illustrates the top level logic of the Explode-Triple procedureof block 16 in FIG. 3. Upon this process being called at start block 21,initialization block 22 sets the incremental production plan to zero andsets the ReqVol ( ) to zero. Then, in function block 23, ShipVol^(I)(d*,td*) is set to IncShipVol and ReqVol (DemRes (d*, td*) is set toIncShipVol. At this point, nested processing block 24 ∀rεAllRes, inorder of decreasing ResIndx(r), and processing block 25 ∀trεAllPer, indecreasing order, are entered. Processing block 25 calls themain-iteration (r, tr) processing block 26. Blocks 24 to 26 show themain iteration (block 26), as contained within the loop through allperiods (block 25), as contained with a loop through all resources(block 24). When processing in block 24 and 25 are completed, a returnis made to the main program shown in FIG. 3.

FIG. 5 illustrates the main iteration 26 of the Explode-Triple procedureof FIG. 4, treating it as a separate procedure for illustrationpurposes. When called at start block 28, a determination is made indecision block 29 as to whether ReqVol (r, tr)=0. If so, a return ismade at block 30 to processing block 26; otherwise, net requirement isnetted against supply in function block 31. Then, in decision block 32 adetermination is made as to whether ReqVol (r, tr)=0. If so, a return ismade at block 33 to processing block 26; otherwise, processing block 34∀jεProdOpns (r) is entered. The requirement to production is exploded byoperation j in function block 35. A determination is made in decisionblock 36 as to whether ReqVol (r, tr)=0. If so, a return is made atblock 37 to processing block 26; otherwise, the processing in processingblock 34 continues to the next j in function block 38. Blocks 29, 30,32, 33, 36, and 37 indicate the various opportunities for the iterationto terminate early. Blocks 34 to 38 indicate the loop through alloperations that produce the resource. Blocks 31, 35 and 39 show how thethree methods for fulfilling requirements fit into the main iteration.When the processing in block 34 completes, the requirement is explodedto stock in period t−1 in function block 39, and a return is made atblock 40 to processing block 26.

The Explode Triple procedure is similar to the explosion technique ofMaterial Requirements Planning. The explosion functions by associating aquantity ReqVol (r, tr) (“requirement volume”) with each resource r ineach period tr. The Explode Triple procedure calls three moreprocedures, Net-Against-Supply, Explode-To-Production, andExplode-To-Stock. These four procedures can be described as follows:

Procedure Explode Triple:

Given: A triple (d*, td*, IncShipVol)εShipSeq

∀dεAllDem, ∀tdεAllPer:

-   -   ShipVol^(I)(d, td)←0

∀jεAllOpn, ∀tjεAllPer:

-   -   ProdVol^(I)(j, tj)←0

∀rεAllRes, ∀trεAllPer:

-   -   StockVol^(I)(r, tr)←0    -   ConsSupVol^(I)(r, tr)←0    -   ReqVol (r, tr)←0

ShipVol^(I)(d*, td*)←IncShipVol

ReqVol (DemRes (d*), td*)←IncShipVol

Do the following for each resource rεAllRes in order of decreasingResIdx (r):

Do the following main iteration for each period trεAllPer in decreasingorder:

-   -   If ReqVol (r, tr)=0, skip this main iteration.    -   Call Net-Against-Supply (r, tr).    -   If ReqVol (r, tr)=0, terminate this main iteration.    -   Do the following for each operation jεProdOpns (r):        -   Call Explode-To-Production (r, j, tr).        -   If ReqVol (r, tr)=0, terminate this main iteration.    -   Call Explode-To-Stock (r, tr).

Return. Procedure Net-Against-Supply:

[Net the requirement against consumed supply.]Given: rεAllRes, trεAllPer.ConsSupVol^(I)(r, tr)←Min ((ConsSupVol (r, tr)−ConsSupVol^(R)(r, tr)),ReqVol (r, tr)) ReqVol (r, tr)←+−ConsSupVol^(I)(r, tr)

Return. Procedure Explode-To-Production:

[Explode the requirement to production by operation j.]Given: rεAllRes, jεProdOpns (r), trεAllPer.ProdVol^(I)(j, tr)←Min ((ProdVol (i, tr)−ProdVoI^(R)(j, tr)), ReqVol (r,tr)) ∀r′εBOR (j):

∀tr′εAllPer:

-   -   ReqVol (r′, tr′)←+UsageRate (j, tr, r′, tr′)*ProdVol^(I)(j, tr)        ReqVol (r, tr)←+−ProdVol^(I)(j, tr)

Return. Procedure Explode-To-Stock:

[Explode the requirement to stock from period tr−1.]Given: rεAllRes, trεAllPer.If tr=1, return.

-   StockVol^(I)(r, tr−1)←Min ((StockVol (r, tr−1)−StockVol^(R)(r,    tr−1)), ReqVol(r, tr))    ReqVol (r, tr−1)←+StockVol^(I)(r, tr−1)    ReqVol (r, tr)←+−StockVol^(I)(r, tr−1)

Return.

It is easy to see that Explode Triple computes an incremental productionplan that satisfies conditions (5), (6), (7), and (8) above. It can alsobe shown that condition (9) is satisfied, by verifying that ReqVol(r,tr)=0 at the end of each main iteration. In MRP terms, this means thatthere are no “net requirements”. In fact, there are a number of majordifferences between the Explode Triple procedure and a conventional MRPexplosion:

-   -   Explode Triple never has any net requirements.    -   Explode Triple can explode a requirement to multiple alternative        operations for producing the same product.    -   Explode Triple can explode a requirement to production in an        earlier period (by exploding to stock).        All of these differences are enabled by exploiting knowledge of        the pre-existing production plan.

The main iteration of Explode Triple fills the requirement, ReqVol (r,tr), using three alternative procedures: Net-Against-Supply,Explode-To-Production, and Explode-To-Stock. The order in which thesethree procedures are called is arbitrary: they can actually be called inany order. Also, the calls to Explode-To-Production for each operationin ProdOpns (r) can be done in any order, and indeed, these calls can beinterlaced with the calls to Net-Against-Supply and Explode-To-Stock inany order. In general, using different alternative orderings for theseprocedure calls will result in different but equally valid peggings.Using any such ordering is considered to be a version of the PBEtechnique.

Depending on the application, there may be business reasons for choosingone ordering instead of another. For example, the ordering used in thealgorithm description above has the effect of pegging production in thelatest possible period to the current shipment. Alternatively, callingExplode-To-Stock before Explode-To-Production would have the effect ofpegging production in the earliest possible period to the currentshipment.

Extension to the General RCPP Problem

The general RCPP problem is allowed to have negative consumption.Unfortunately, when negative is consumption is allowed, it is easy toconstruct examples of RCPP problems and production plans such that nofeasible partitioning of the production plan exists. Thus the PBEtechnique for PC RCPP problems, which is designed to construct afeasible partitioning, cannot be expected to function properly in thepresence of negative consumption.

Given an RCPP problem with negative consumption and a correspondingfeasible production plan, consider the following definitions:

∀j ∈ AllOpn, ∀tj ∈ AllPer, ∀r ∈ BOR(j), ∀tr ∈ AllPer:$\quad\mspace{79mu} {{{If}\mspace{14mu} {{UsageRate}\left( {j,{tj},r,{tr}} \right)}} < {0\text{:}{\quad\mspace{130mu} {{{{Let}\mspace{14mu} {NegUsageRate}\; \left( {j,{tj},r,{tr}} \right)} = {{{{- {UsageRate}}\; \left( {j,{tj},r,{tr}} \right)\mspace{79mu} {If}\mspace{14mu} {UsageRate}\; \left( {j,{tj},r,{tr}} \right)} \geq {0\text{:}\mspace{130mu} {Let}\mspace{14mu} {NegUsageRate}\; \left( {j,{tj},r,{tr}} \right)}} = {0{\forall{r \in {AllRes}}}}}},{{\forall{{tr} \in {{AllPer}\text{:}{Let}\mspace{14mu} {NegConsVol}\; \left( {r,{tr}} \right)}}} = {\underset{{tj} \in \; {AllPers}}{\sum\limits_{j \in \; {{ConsOpns}{(r)}}}}\; {{NegUsageRate}\; \left( {j,{tj},r,{tr}} \right)*{{ProdVol}\left( {j,{tj}} \right)}}}}}}}}$

Thus, NegConsVol (r, tr) is the total negative consumption of resource rin period tr.

To see how the PBE technique is extended to handle the negativeconsumption case, define the auxiliary problem and auxiliary productionplan, by making the following modifications to the original RCPP problemand production plan:

∀rεAllRes:

Add an “auxiliary” operation, j*.

Let ProdRes (j*)=r.

Let BOR (j*)=Ø.

∀trεAllPer:

-   -   Let ProdVol (j*, tr)=NegConsVol (r, tr).        ∀jεAllOpn, ∀tjεAllPer, ∀rεBOR (j), ∀trεAllPer:

If UsageRate (j, tj, r, tr)<0:

-   -   UsageRate (j, tj, r, tr)←0        In other words, the negative consumption is replaced by        production from auxiliary operations and the negative usage        rates are set to zero. The auxiliary problem and production plan        have two important properties:    -   The auxiliary problem is a Positive Consumption RCPP problem.    -   The auxiliary production plan is feasible for the auxiliary        problem.        The PBE technique can be extended to handle negative consumption        in either of the following two ways:    -   Explicit transformation: Explicitly construct the auxiliary        problem and production plan, apply the PBE technique to them,        and then transfer the resulting pegging back into the original        problem and production plan.    -   Implicit transformation: Work with the original problem and        production plan, by adding steps to the PBE technique that are        mathematically equivalent to what the PBE technique would be        doing with the auxiliary problem and production plan.        More specifically, the implicit transformation approach makes        the following modifications to the original algorithm. In the        Explode-To-Production procedure, any value of UsageRate(j, tr,        r′, tr′) that is negative is treated as zero. In addition to        pegging the production volumes, etc., it also explicitly pegs        the negative consumption volumes, by working with PgdNegConsVol        (r, tr, d, td), NegConsVol^(R)(r, tr), and NegConsVol^(I)(r,        tr). A call to the following procedure is added (in any order)        to the Explode-For-PBE procedure:

Procedure Explode-To-Negative-Consumption:

Given: rεAllRes, trεAllPer.NegConsVol^(I)(r, tr)←Min ((NegConsVol (r, tr)−NegConsVol^(R)(r, tr)),ReqVol (r, tr)) ReqVol (r, tr)←+−NegConsVol^(I)(r, tr)

Return.

The pegging constructed by the PBE technique in the negative consumptioncase is a feasible partitioning of the auxiliary production plan withrespect to the auxiliary problem. This means that the pegging haspartitioned the original production plan into a set of disjoint portionsassociated with each shipment, where each portion defines a feasibleproduction plan for achieving its corresponding shipment, if negativeconsumption is treated as production by an auxiliary operation. This isa precise way of saying that the pegging is plausible in the presence ofnegative consumption.

Substitution

The definition of the RCPP problem can be extended to includesubstitution in the BOR structure. Specifically, given any operations,for any resource rεBOR (j), an RCPP problem with substitution mayspecify a set of substitute resources, any of which can be consumed inplace of r, and with potentially different usage rates, etc. For adetailed formulation of the RCPP problem with substitution, see U.S.Pat. No. 5,630,070.

Any RCPP problem with substitution can be transformed into an equivalentRCPP problem without substitution by the following technique. For eachoperation j and each resource rεBOR (j) that has substitution, modify tothe problem as follows:

Add an extra capacity resource, r*.

Add r* to BOR (j).

Add an extra operation, j*.Let ProdRes (j*)=r*.

Let BOR (*)=r.

For each substitute resource r′ associated with j and r, do thefollowing:

Add an extra operation, j**

Let ProdRes (j**)=r*.

Let BOR (j**)=r′.

Remove r from BOR (j).

Also, the usage rate data for the new objects is copied from the usagerates in the original data as appropriate. This transformed problem isequivalent to the original problem in the sense that it leads to anequivalent set of feasible production plans. Also, the transformedproblem has no substitution. The PBE technique can be extended to handlesubstitution either by using an explicit transformation approach or byusing an implicit transformation approach, with respect to thetransformation defined above.

Pegging Infeasibilities

Some RCPP systems allow the production plan that they produce to beinfeasible. In particular, the resource constraints are sometimesallowed be to violated, perhaps with a penalty in an objective function.The PBE technique can be extended to cover this case, by using a problemreduction approach. Given an RCPP problem and a production plan thatviolates some of the resource constraints, but is otherwise feasible,define the following transformed problem. For each resource r and eachperiod tr whose resource constraint is violated, modify the problem andproduction plan as follows:

-   Let ViolVol (r, tr)=The amount by which the left-hand side of the    resource constraint on tr and r exceeds the right-hand-side    (“violation volume”).-   Add an extra operation, j*.-   Let ProdRes (j*)=r.-   Let BOR (j*)=Ø.-   Let ProdVol (j*, tr)=ViolVol (r, tr).    It is easy to see that the transformed production plan is feasible    for the transformed problem.

This transformation can be used to extend the PBE technique to handleresource constraint violations by applying either an explicittransformation approach or an implicit transformation approach. Ineither case, the production pegging for the extra operations has specialsignificance. For any demand d and any period td, PgdProdVol (j*, tr, d,td) is the amount of the violation of the resource constraint forresource r in period tr that is pegged to the shipment to demand d inperiod td. (It might be notated as PgdViolVol (r, tr, d, td).)

Note that, when using the implicit approach, the use of ViolVol (r, tr)to fill the requirement ReqVol (r, tr) in the Explode-For-PBE procedurecan be done in any order compared to the other procedures for fillingthe requirement (Net-To-Supply, Explode-To-Production, Explode-To-Stock,and Explode-To-Negative-Consumption). Normally, it would be done last,since violating a resource constraint would normally be done only as alast resort. A special case of RCPP where resource constraints can beviolated is MRP. Thus the above techniques can be used to do pegging ofa pre-existing MRP solution.

While the invention has been described in terms of preferredembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theappended claims.

1. A computer implemented method for constructing a pegging of aproduction plan for a resource-constrained production planning (RCPP)problem, comprising the step of constructing the pegging after theproduction plan has already been generated.
 2. The computer implementedmethod recited in claim 1, wherein the step of constructing the peggingcomprises the steps of: reconstructing a modified version of theproduction plan by an incremental allocation technique, and thenapplying a Pegging By Monitoring technique to the reconstructedproduction plan in order to produce the pegging.
 3. The computerimplemented method recited in claim 2, wherein the step ofreconstructing a modified verison of the production plan is accomplishedby a series of “explosion” steps similar to an explosion technique ofMaterial Requirements Planning (MRP).
 4. The computer implemented methodrecited in claim 3, wherein the method is restricted to be used only onPositive Consumption resource-constrained production planning (RCPP)problems and comprises of the steps of: reading in a PositiveConsumption RCPP problem, reading in a feasible production plan for theproblem, either reading in a shipment sequence corresponding to theshipments in the production plan, or computing a default shipmentsequence, initializing the reconstructed production plan to zero,initializing the pegging to zero, proceeding through the triples (d*,td*, IncShipVol) in the shipment sequence, with each demand d* in periodtd*, and performing the following three steps: employing a specializedexplosion technique to construct an incremental production plan (denotedby ProdVol^(I)( ), StockVol^(I)( ), ConsSupVol^(I)( ), and ShipVol^(I)()) that specifies a feasible means of enabling a shipment of IncShipVolunits to demand d* in period td* without using any resources that havealready been pegged, adding the incremental production plan to thereconstructed production plan, and pegging the incremental productionplan to the current shipment (d*, td*), and reporting the pegging. 5.The computer implemented method recited in claim 4, wherein thespecialized explosion technique comprises the steps of: initializing theincremental production plan to zero, associating a quantity, ReqVol (r,tr), with each resource r in each period tr, initializing all values ofReqVol (r, tr) to zero, setting ShipVol^(I)(d*, td*) to IncShipVol,setting ReqVol (DemRes (d*), td*) to IncShipVol, proceeding through eachresource rεAllRes in order of decreasing ResIdx (r) and performing thefollowing five steps: proceeding through each period trεAllPer indecreasing order and performing the following four steps: netting therequirement against supply, ∀jεProdOpns (r), performing the followingstep: exploding the requirement to production by operation j, andexploding the requirement to stock in period tr−1.
 6. The computerimplemented method recited in claim 5, wherein the step netting therequirement against supply comprises the steps of: settingConsSupVol^(I)(r, tr) Min ((ConsSupVol (r, tr)−ConsSupVol^(R)(r, tr)),ReqVol (r, tr)), and setting ReqVol (r, tr) ←+−ConsSupVol^(I)(r, tr),and wherein the step exploding the requirement to production byoperation j comprises the steps of: setting ProdVol^(I)(j, tr)←Min((ProdVol (j, tr)−ProdVol^(R)(j, tr)), ReqVol (r, tr)), ∀r′εBOR (j),where BOR is Bill-of-Resources, performing the following three steps:∀tr′εAllPer, performing the following step: setting ReqVol (r′,tr′)+UsageRate (j, tr, r′, tr′) * ProdVol^(I)(j, tr), setting ReqVol (r,tr)←+−ProdVol^(I)(j, tr), and wherein the step exploding the requirementto stock in period tr−1 comprises the steps of: if tr=1, doing nothing,but otherwise performing the following three steps: settingStockVol^(I)(r, tr−1)←Min ((StockVol (r, tr−1)−StockVol^(R)(r, tr−1)),ReqVol (r, tr)), setting ReqVol (r, tr−1)←+StockVol^(I)(r, tr−1), andsetting ReqVol (r, tr)←+−StockVol^(I)(r, tr−1).
 7. The computerimplemented method recited in claim 6, wherein the steps netting therequirement against supply, exploding the requirement to production byoperation j, and exploding the requirement to stock in period tr−1 areperformed in any order, where this method is called Pegging By ExplosionTechnique for Positive Consumption RCPP Problems, or PC-PBE technique.8. The computer implemented method recited in claim 7, (the PC-PCEtechnique), used to construct a pegging of a production plan for anyRCPP problem, by using the method within the steps of: explicitlyconstructing an auxiliary problem and production plan identical to theoriginal problem and production plan, except that for each resource, an“auxiliary” operation is added that produces that resource and consumesno resources, the negative consumption of each resource is replaced withproduction by the auxiliary operation for that resource, and allnegative usage rates are set to zero, and applying the PC-PBE techniqueto the auxiliary problem and production plan, and transferring theresulting pegging back into the original problem and production plan. 9.The computer implemented method recited in claim 7, (the PC-PBEtechnique), used to construct a pegging of a production plan for anyRCPP problem, by using the method within the steps of: conceptuallydefining the auxiliary problem and production plan identical to theoriginal problem and production plan, except that for each resource, an“auxiliary” operation is added that produces that resource and consumesno resources, the negative consumption of each resource is replaced withproduction by the auxiliary operation for that resource, and allnegative usage rates are set to zero. working with the original problemand production plan, and performing a sequence of steps that aremathematically equivalent to those that the PC-PBE technique would beperforming, if it were being applied to the auxiliary problem andproduction plan, whereby the method is called Pegging By ExplosionTechnique for RCPP Problems, or the PBE technique.
 10. The computerimplemented method recited in claim 9, (the PBE technique), used toconstruct a pegging of a production plan for a variant of an RCPPproblem that has been generalized so as to allow substitution in a Billof Resources (BOR) structure, by using the method within the steps of:conceptually defining a transformed RCPP problem and production planthat is equivalent to the original problem and production plan, but hasno substitution, working with the original problem and production plan,and performing a sequence of steps that are mathematically equivalent tothose that the PBE technique would be performing, if it were beingapplied to the transformed problem and production plan.
 11. The computerimplemented method recited in claim 9, (the PBE technique), used toconstruct a pegging of a production plan for a variant of an RCPPproblem that has been generalized so as to allow, substitution in aBill-of-Resources (BOR) structure and to permit violations of theresource constraints, by using the method within the steps of:conceptually defining a transformed RCPP problem and production planthat is equivalent to the original problem and production plan, but hasno substitution and does not violate any of the resource constraints,working with the original problem and production plan, and performing asequence of steps that are mathematically equivalent to those that thePBE technique would be performing, if it were being applied to thetransformed problem and production plan.
 12. The computer implementedmethod recited in claim 11, specifically applied to construct a peggingof the solution of a Material Requirements Planning problem.
 13. Thecomputer implemented method recited in claim 7, wherein the method isrestricted to be used only on problems in which Nperiods=1, where themethod is made simpler to implement in the following ways: byeliminating the concept of a period from all of the data and notation;by eliminating the distinction between material resources and capacityresources; by deleting all references to stock volume (StockVol ( )) andits pegging; and by removing the exploding the requirement to stock stepfrom the method.
 14. The computer implemented method recited in claim 7,wherein the method is restricted to be used only on problems in whicheach resource can only be produced by at most one operation, where themethod is made simpler to implement in the following ways: byeliminating the concept of an operation from all of the data andnotation, eliminating all data associated with an operation andreplacing it by corresponding data associated with a product, so that,e.g., ProdVol (, tj) for operation j is replaced by ProdVol (r, tr) forproduct resource r; by eliminating the following two steps from method:∀jεProdOpns (r), performing the following step: exploding therequirement to production by operation j; by replacing the above twosteps with the following step: exploding the requirement to production,if the resource is a product.
 15. The computer implemented methodrecited in claim 7, wherein the method is restricted to be used only onproblems in which Nperiods=1 and in which each resource can only beproduced by at most one operation, where the method is made simpler toimplement in the following ways: by eliminating the concept of a periodfrom all of the data and notation; by eliminating the distinctionbetween material resources and capacity resources; by deleting allreferences to stock volume (StockVol 0) and its pegging; by removing theexploding the requirement to stock step from the method. by eliminatingthe concept of an operation from all of the data and notation,eliminating all data associated with an operation and replacing it bycorresponding data associated with a product, so that, e.g., ProdVol (j,tj) for operation j is replaced by ProdVol (r, tr) for product resourcer; by eliminating the following two steps from method: ∀jεProdOpns (r),performing the following step: exploding the requirement to productionby operation j; by replacing the above two steps with the followingstep: exploding the requirement to production, if the resource is aproduct.
 16. A computer readable medium containing code implementing amethod for constructing a pegging of a production plan for aresource-constrained production planning (RCPP) problem, said methodcomprising the step of constructing the pegging after the productionplan has already been generated.
 17. The computer readable mediumrecited in claim 16, wherein the step of constructing the peggingcomprises the steps of: reconstructing a modified version of theproduction plan by an incremental allocation technique, and thenapplying a Pegging By Monitoring technique to the reconstructedproduction plan in order to produce the pegging.
 18. The computerreadable medium recited in claim 17, wherein the step of reconstructinga modified verison of the production plan is accomplished by a series of“explosion” steps similar to an explosion technique of MaterialRequirements Planning (MRP).
 19. The computer readable medium recited inclaim 18, wherein the method is restricted to be used only on PositiveConsumption resource-constrained production planning (RCPP) problems andcomprises of the steps of: reading in a Positive Consumption RCPPproblem, reading in a feasible production plan for the problem, eitherreading in a shipment sequence corresponding to the shipments in theproduction plan, or computing a default shipment sequence, initializingthe reconstructed production plan to zero, initializing the pegging tozero, proceeding through the triples (d*, td*, IncShipVol) in theshipment sequence, with each demand d* in period td*, and performing thefollowing three steps: employing a specialized explosion technique toconstruct an incremental production plan (denoted by ProdVol^(I)( ),StockVol^(I)( ), ConsSupVol^(I)( ), and ShipVol^(I)( )) that specifies afeasible means of enabling a shipment of IncShipVol units to demand d*in period td* without using any resources that have already been pegged,adding the incremental production plan to the reconstructed productionplan, and pegging the incremental production plan to the currentshipment (d*, td*), and reporting the pegging.
 20. The computer readablemedium recited in claim 19, wherein the specialized explosion techniquecomprises the steps of: initializing the incremental production plan tozero, associating a quantity, ReqVol (r, tr), with each resource r ineach period tr, initializing all values of ReqVol (r, tr) to zero,setting ShipVol^(I)(d*, td*) to IncShipVol, setting ReqVol (DemRes (d*),td*) to IncShipVol, proceeding through each resource r C AllRes in orderof decreasing ResIdx (r) and performing the following five steps:proceeding through each period trεAllPer in decreasing order andperforming the following four steps: netting the requirement againstsupply, ∀jεProdOpns (r), performing the following step: exploding therequirement to production by operation j, and exploding the requirementto stock in period tr−1.
 21. The computer readable medium recited inclaim 20, wherein the step netting the requirement against supplycomprises the steps of: setting ConsSupVol^(I)(r, tr) Min ((ConsSupVol(r, tr)−ConsSupVol^(R)(r, tr)), ReqVol (r, tr)), and setting ReqVol (r,tr)←+−ConsSupVol^(I)(r, tr), and wherein the step exploding therequirement to production by operation j comprises the steps of: settingProdVol^(I)(j, tr) Min ((ProdVol (j, tr)−ProdVol^(R)(j, tr)), ReqVol (r,tr)), ∀r′εBOR (j), where BOR is Bill-of-Resources, performing thefollowing three steps: ∀tr′εAllPer, performing the following step:setting ReqVol (r′, tr′)+UsageRate (j, tr, r′, tr′)* ProdVol^(I)(j, tr),setting ReqVol (r, tr)←+−ProdVol^(I)(j, tr), and wherein the stepexploding the requirement to stock in period tr−1 comprises the stepsof: if tr=1, doing nothing, but otherwise performing the following threesteps: setting StockVol^(I)(r, tr−1)←Min ((StockVol (r,tr−1)−StockVol^(R)(r, tr−1)), ReqVol (r, tr)), setting ReqVol (r,tr−1)←+StockVol^(I)(r, tr−1), and setting ReqVol (r,tr)←+−StockVol^(I)(r, tr−1).
 22. The computer readable medium recited inclaim 20, wherein the steps netting the requirement against supply,exploding the requirement to production by operation j, and explodingthe requirement to stock in period tr−1 are performed in any order,where this method is called Pegging By Explosion Technique for PositiveConsumption RCPP Problems, or PC-PBE technique.
 23. The computerreadable medium recited in claim 22, (the code implementing the PC-PBEtechnique), where in the method is used to construct a pegging of aproduction plan for any RCPP problem by using the method within thesteps of: explicitly constructing an auxiliary problem and productionplan identical to the original problem and production plan, except thatfor each resource, an “auxiliary” operation is added that produces thatresource and consumes no resources, the negative consumption of eachresource is replaced with production by the auxiliary operation for thatresource, and all negative usage rates are set to zero, and applying thePC-PBE technique to the auxiliary problem and production plan, andtransferring the resulting pegging back into the original problem andproduction plan.
 24. The computer readable medium recited in claim 22,(the code implementing the PC-PBE technique), wherein the method is usedto construct a pegging of a production plan for any RCPP problem, byusing the method within the steps of: conceptually defining theauxiliary problem and production plan identical to the original problemand production plan, except that for each resource, an “auxiliary”operation is added that produces that resource and consumes noresources, the negative consumption of each resource is replaced withproduction by the auxiliary operation for that resource, and allnegative usage rates are set to zero, working with the original problemand production plan, and performing a sequence of steps that aremathematically equivalent to those that the PC-PBE technique would beperforming, if it were being applied to the auxiliary problem andproduction plan, whereby the method is called Pegging By ExplosionTechnique for RCPP Problems, or the PBE technique.
 25. The computerreadable medium, recited in claim 24, (the code implementing the PBEtechnique), wherein the method is used to contruct a pegging of aproduction plan for a variant of an ECPP problem that has beengeneralized so as to allow substitution in a Bill-of-Resources (BOR)structure, by using the method within the steps of: conceptuallydefining a transformed RCPP problem and production plan that isequivalent to the original problem and production plan, but has nosubstitution, working with the original problem and production plan, andperforming a sequence of steps that are mathematically equivalent tothose that the PBE technique would be performing, if it were beingapplied to the transformed problem and production plan.
 26. The computerreadable medium recited in claim 24, (the code implementing the PBEtechnique), wherein the method is used to construct a pegging of aproduction plan for a variant of an RCPP problem that has beengeneralized so as to allow substitution in a Bill-of-Resources (BOR)structure and to permit violations of the resource constraints, by usingthe method within the steps of: conceptually defining a transformed RCPPproblem and production plan that is equivalent to the original problemand production plan, but has no substitution and does not violate any ofthe resource constraints, working with the original problem andproduction plan, and performing a sequence of steps that aremathematically equivalent to those that the PBE technique would beperforming, if it were being applied to the transformed problem andproduction plan.
 27. The computer readable medium recited in claim 26,specifically applied to construct a pegging of the solution of aMaterial Requirements Planning problem.
 28. The computer readable mediumrecited in claim 22, wherein the method is restricted to be used only onproblems in which Nperiods=1, where the method is made simpler toimplement in the following ways: by eliminating the concept of a periodfrom all of the data and notation; by eliminating the distinctionbetween material resources and capacity resources; by deleting allreferences to stock volume (StockVol 0) and its pegging; and by removingthe exploding the requirement to stock step from the method.
 29. Thecomputer readable medium recited in claim 22, wherein the method isrestricted to be used only on problems in which each resource can onlybe produced by at most one operation, where the method is made simplerto implement in the following ways: by eliminating the concept of anoperation from all of the data and notation, eliminating all dataassociated with an operation and replacing it by corresponding dataassociated with a product, so that, e.g., ProdVol (j, tj) for operationj is replaced by ProdVol (r, tr) for product resource r; by eliminatingthe following two steps from method: ∀jεProdOpns (r), performing thefollowing step: exploding the requirement to production by operation j;by replacing the above two steps with the following step: exploding therequirement to production, if the resource is a product.
 30. Thecomputer readable medium recited in claim 22, wherein the method isrestricted to be used only on problems in which Nperiods=1 and in whicheach resource can only be produced by at most one operation, where themethod is made simpler to implement in the following ways: byeliminating the concept of a period from all of the data and notation;by eliminating the distinction between material resources and capacityresources; by deleting all references to stock volume (StockVol 0) andits pegging; by removing the exploding the requirement to stock stepfrom the method. by eliminating the concept of an operation from all ofthe data and notation, eliminating all data associated with an operationand replacing it by corresponding data associated with a product, sothat, e.g., ProdVol (j, tj) for operation j is replaced by ProdVol (r,tr) for product resource r; by eliminating the following two steps frommethod: ∀jεProdOpns (r), performing the following step: exploding therequirement to production by operation j; by replacing the above twosteps with the following step: exploding the requirement to production,if the resource is a product.